(Note: the essence of this answer is just the first three paragraphs. Everything below the line is just to try to reconcile the answer with the answer linked in the question, so feel free to not read that part if you're short on time.)
In Chapter Twenty-Three of Half-Blood Prince Dumbledore explicitly refers to the diary as Voldemort's first Horcrux (my emphasis).
"The careless way in which Voldemort regarded this Horcrux seemed most ominous to me. It suggested that he must have made — or been planning to make — more Horcruxes, so that the loss of his first would not be so detrimental. I did not wish to believe it, but nothing else seemed to make sense.
We are not told how Dumbledore knew that the diary was the first one, and it is theoretically possible that Dumbledore was using the term "first" imprecisely, but nonetheless this is what he said.
As for the argument in the linked answer that the ring was the first Horcrux, we can simply say that (as the answer itself seems to admit) it is possible that Voldemort waited a while between retrieving the ring and making it into a Horcrux.
In fact, we could even argue that Voldemort did not know about Horcruxes at the time he stole the ring. If, as per the linked answer, Voldemort was only 15 at the time he stole it, it would have been the summer between his fourth and fifth years. In Half-Blood Prince in the memory in which Voldemort asks Slughorn about Horcruxes, Voldemort had already committed the murders and stolen the ring:
Harry recognized Voldemort at once. His was the most handsome face and he looked the most relaxed of all the boys. His right hand lay negligently upon the arm off his chair; with a jolt, Harry saw that he was wearing Marvolo's gold-and-black ring; he had already killed his father.
The question is what year did this event occur in? We can probably safely assume that it was not Voldemort's seventh year, because Slughorn refers to him as a prefect rather than as the Head Boy, which he would have been if it was his seventh year:
"You don't want to be caught out of bed out of hours, and you a prefect..."
So it must have been his fifth or sixth year. Now his fifth year was the year the Chamber of Secrets was opened. We know that until Hagrid was caught it was considered dangerous to be in the corridors at night, as Dumbledore stated in Riddle's memory in Chamber of Secrets:
"Well, hurry off to bed," said Dumbledore, giving Riddle exactly the kind of penetrating stare harry knew so well. "Best not to roam the corridors these days. Not since..."
It thus seems unlikely that Slughorn's party would have been during the time the monster was at large, as we know the party ended rather late:
Bewildered, Harry looked around as a small golden clock standing upon Slughorn's desk chimed eleven o'clock.
"Good gracious, is it that time already?" said Slughorn. "You'd better get going boys, or we'll all be in trouble.
It seems that Slughorn was only concerned about the general issue of students in the corridors at night being not allowed, and not the bigger issue of the danger of the monster on the loose. Thus, this may indicate that this party took place at a time when the Chamber of Secrets was not an issue. We know the date that Voldemort caught Hagrid, because it was mentioned in the diary:
Mouth hanging open, Harry saw that the little square for June thirteenth seemed to have turned into a minuscule television screen.
So based on this we might conclude that this party was either in Voldemort's sixth year, or at the very end (after June thirteenth) of his fifth year, or much earlier in his fifth year before the Chamber of Secrets became an issue. There is a possible indication that this party took place later in the year rather than earlier. During the party, Voldemort asked the following question:
"Sir, is it true that Professor Merrythought is retiring?" he asked.
It would seem more likely to ask this at the end of a year then at the beginning of a year. Furthermore, we actually know that Professor Merrythought was still teaching through the end of Voldemort's education. Dumbledore tells Harry later in Half-blood Prince that Voldemort asked for a teaching job after he graduated. When Harry asked which subject, Dumbledore said:
"Defense Against the Dark Arts. It was taught at the time by an old Professor by the name of Galatea Merrythought, who had been at Hogwarts for nearly fifty years.
This implies that Merrythought didn't retire until after Voldemort's seventh year (and his retirement then was perhaps what opened up the position). It would make more sense that Voldemort would mention the rumor of Merrythought's retirement in his sixth year than in his fifth year, because in his fifth year it would have been more than two years before Merrythought actually retired.
If we then assume that this party occurred during his sixth year, and towards the end of the year at that, it would mean that this was approaching two years since he had stolen the ring. If Voldemort had known about Horcruxes at the time when he sole the ring, why did it take him so long to ask Slughorn about them? It would make more sense if Voldemort had only recently found out about Horcruxes, and now he was asking Slughorn at (one of) the first opportunity(ies). Thus, even if Voldemort had the ring before the events involving the diary and the Chamber of Secrets, it is possible that he did not become aware of Horcuxes until later, and once he became aware he made the diary into the first Horcrux.
A small issue with this is that it seems that Riddle enchanted the diary at the latest in the beginning of his sixth year, based on what his memory told Harry at the end of Chamber of Secrets:
I decided to leave behind a diary preserving my sixteen-year-old self in its pages, so that one day, with luck, I would be able to lead another in my footsteps, and finish Salazar Slytherin's noble work."
This implies that he was still sixteen at the time, and as he would have turned seventeen a few months into his sixth year, this would seem to indicate that he had already enchanted the diary prior to the estimated time I gave for Slughorn's party, which would mean that he did know about Horcruxes for a while before asking Slughorn about them.
However, there are several possible ways to deal with this. We could suggest that Voldemort in fact enchanted the diary later (e.g. when he was seventeen) but preserved his "sixteen-year-old self" because that was who had opened the Chamber of Secrets. Alternatively, it is possible that there were two levels of enchantment on the diary. Perhaps at sixteen he simply implanted the memory, hoping to be able to use that to show subsequent students how to open the Chamber of Secrets. But when he later found out about Horcruxes he improved the diary by actually placing part of his soul in it, which would then give him the ability to reopen the Chamber of secrets via possessing a future student rather than relying on a memory alone. Or we could play with the dates a little and say that Voldemort found out about Horcruxes shortly before his seventeenth birthday and he created the diary Horcrux then, and it took him a little while to get the right opportunity to talk to Slughorn, such that the party could still have been well into his sixth year (e.g. he made the Horcrux in December and the party was in March).
Whichever route we choose leaves open the possibility that Voldemort had not yet known about Horcruxes when he stole the ring, which means that it is possible that he had the ring before the diary yet did not make it into a Horcrux until after the diary.